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A die is thrown until the first time the total sum of the face values of the die is 700 or greater. Estimate the probability that, for this to happen,

  1. more than 210 tosses are required
  2. less than 190 tosses are required
  3. between 180 and 210 tosses, inclusive, are required

Solution

First, we will find the expected value and variance.

\[ E[X] = \frac{\text{sum of possible outcomes}}{\text{number of outcomes}} = \frac{1+2+3+4+5+6}{6}=\frac{7}{2} \approx 3.5\\ Var[X] = E[X^2] - (E[X])^2\\ E[X^2] = \frac{1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2}{6}= \frac{91}{6}\\ Var[X] = \frac{91}{6} - (\frac{7}{2})^2 = \frac{35}{12} \approx 2.91667\\ \]

outcomes <- c(1,2,3,4,5,6)

sum_outcomes <- sum(outcomes)
count_outcomes <- length(outcomes)

EX <- sum_outcomes / count_outcomes
print(EX)
## [1] 3.5
outcomes_sq <- c(1^2,2^2,3^2,4^2,5^2,6^2)
sum_outcomes_sq <- sum(outcomes_sq)
EX_sq <- sum_outcomes_sq/count_outcomes
varx <- EX_sq - (EX)^2
print(varx)
## [1] 2.916667

The sum of each outcome is independent and identically distributed.

\[ E[S_N] = E[\Sigma_{i=1}^{N}X[i]] = \Sigma_{i=1}^N E[X[i]] = \Sigma_{i=1}^N = \frac{7}{2}N\\ Var[S_N] = Var(\Sigma_{i=1}^N X[i]) = \Sigma_{i=1}^N Var[X[i]] = \Sigma_{i=1}^N \frac{35}{12} = \frac{35}{12}N\\ \]

more than 210 tosses are required.

\[ P(S_{210} < 700)\\ E[S_{210}] = \frac{7}{2}\cdot 210 = 735\\ Var[S_{210}] = \frac{35}{12}\cdot 210 = 612.5\\ P(S_{210}<700)= P(\frac{S_{210}-E[S_{210}]}{\sqrt{Var[S_{210}]}} < \frac{700 - E[S_{210}]}{\sqrt{Var[S_{210}]}})\\ =P(Z<\frac{700-735}{\sqrt{612.5}}) \approx P(Z<-1.414) \approx 0.0787 \]

z1 <- ((700-735)/sqrt(612.5))
x1 <- pnorm(z1)
print(x1)
## [1] 0.0786496

less than 190 tosses are required.

\[ P(S_{190}\geq700)\\ E[S_{190}] = \frac{7}{2}\cdot 190 = 665\\ Var[S_{190}] = \frac{35}{12}\cdot 190 = \frac{3325}{6} \approx 554.167\\ P(S_{190}\geq700) = P(\frac{S_{190}-E[S_{190}]}{\sqrt{Var[S_{190}]}} \geq \frac{700 - E[S_{190}]}{\sqrt{Var[S_{190}]}})\\ = P(Z \geq \frac{700-665}{\sqrt{554.167}}) = P(Z \geq 1.48678) = 1-P(Z<1.48678) \approx 0.068536 \]

z2 <- ((700-665)/sqrt(554.167))
x2 <- 1-pnorm(z2)
print(x2)
## [1] 0.06853601

between 180 and 210 tosses, inclusive, are required.

\[ P(S_{180} \leq 700 \leq S_{210}) = P(S_{180}<700)- P(S_{210} < 700) \\ E[S_{180}]=\frac{7}{2}\cdot 180 = 630\\ Var[S_{180}] = \frac{35}{12} \cdot 180 = 525\\ P(S_{180}<700)= P(\frac{S_{180}-E[S_{180}]}{\sqrt{Var[S_{180}]}} < \frac{700-E[S_{180}]}{\sqrt{Var[S_{180}]}})\\ = P(Z<\frac{700-630}{\sqrt{525}}) = P(Z<3.05505) \approx 0.9988749\\ P(S_{180}\leq 700 \leq S_{210}) = P(S_{180} < 700)- P(S_{210}< 700)\\ = P(S_{180}<700) - P(S_{210} < 700))\\ = 0.9988749-0.0787 \approx 0.9202 \]

z3 <- ((700-630)/sqrt(525))
x3 <- pnorm(z3)
print(x3)
## [1] 0.9988749
x3-x1
## [1] 0.9202253